LMFDB - Embedded newform 3344.2.a.v.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3344,2,Mod(1,3344)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3344, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("3344.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3344 = 2^{4} \cdot 11 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3344.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$26.7019744359$
Analytic rank:	$1$
Dimension:	$6$
Coefficient field:	6.6.576096652.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 2x^{5} - 8x^{4} + 11x^{3} + 16x^{2} - 6x - 4 $ x^6 - 2x^5 - 8x^4 + 11x^3 + 16x^2 - 6*x - 4
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1672)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-2.21174$ of defining polynomial
Character	$\chi$	$=$	3344.1

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.21174 q^{3} +2.30747 q^{5} +1.34499 q^{7} +7.31527 q^{9} -1.00000 q^{11} -1.34499 q^{13} -7.41101 q^{15} +0.904266 q^{17} -1.00000 q^{19} -4.31976 q^{21} -0.548933 q^{23} +0.324437 q^{25} -13.8595 q^{27} -4.32803 q^{29} -1.89392 q^{31} +3.21174 q^{33} +3.10353 q^{35} -0.546801 q^{37} +4.31976 q^{39} -4.86420 q^{41} -5.24127 q^{43} +16.8798 q^{45} +5.11346 q^{47} -5.19100 q^{49} -2.90427 q^{51} -3.63971 q^{53} -2.30747 q^{55} +3.21174 q^{57} -13.7786 q^{59} +3.90408 q^{61} +9.83897 q^{63} -3.10353 q^{65} +5.72084 q^{67} +1.76303 q^{69} +4.49402 q^{71} +1.28228 q^{73} -1.04201 q^{75} -1.34499 q^{77} +6.80150 q^{79} +22.5674 q^{81} +7.06560 q^{83} +2.08657 q^{85} +13.9005 q^{87} +15.8923 q^{89} -1.80900 q^{91} +6.08279 q^{93} -2.30747 q^{95} +12.2842 q^{97} -7.31527 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 4 q^{3} + q^{5} + 4 q^{9} - 6 q^{11} - 7 q^{15} + 3 q^{17} - 6 q^{19} + 7 q^{21} - 7 q^{23} + 3 q^{25} - 13 q^{27} - 4 q^{29} - 7 q^{31} + 4 q^{33} - 6 q^{35} - 2 q^{37} - 7 q^{39} + 7 q^{41} - 21 q^{43}+ \cdots - 4 q^{99}+O(q^{100}) $ 6 * q - 4 * q^3 + q^5 + 4 * q^9 - 6 * q^11 - 7 * q^15 + 3 * q^17 - 6 * q^19 + 7 * q^21 - 7 * q^23 + 3 * q^25 - 13 * q^27 - 4 * q^29 - 7 * q^31 + 4 * q^33 - 6 * q^35 - 2 * q^37 - 7 * q^39 + 7 * q^41 - 21 * q^43 + 11 * q^45 - 16 * q^47 + 2 * q^49 - 15 * q^51 + 17 * q^53 - q^55 + 4 * q^57 - 19 * q^59 - 6 * q^61 - 2 * q^63 + 6 * q^65 - 14 * q^67 + q^69 - q^71 - 5 * q^73 - 18 * q^75 + 2 * q^81 - 11 * q^83 - 26 * q^85 - 14 * q^87 + 12 * q^89 - 44 * q^91 - 6 * q^93 - q^95 + 14 * q^97 - 4 * q^99

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	−3.21174	−1.85430	−0.927149	−	0.374692i	$-0.877748\pi$
$3$	−3.21174	−1.85430	−0.927149	+	0.374692i	$0.877748\pi$
$4$	0	0
$5$	2.30747	1.03193	0.515967	−	0.856609i	$-0.327433\pi$
$5$	2.30747	1.03193	0.515967	+	0.856609i	$0.327433\pi$
$6$	0	0
$7$	1.34499	0.508359	0.254179	−	0.967157i	$-0.418195\pi$
$7$	1.34499	0.508359	0.254179	+	0.967157i	$0.418195\pi$
$8$	0	0
$9$	7.31527	2.43842
$10$	0	0
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	−1.34499	−0.373033	−0.186517	−	0.982452i	$-0.559720\pi$
$13$	−1.34499	−0.373033	−0.186517	+	0.982452i	$0.559720\pi$
$14$	0	0
$15$	−7.41101	−1.91351
$16$	0	0
$17$	0.904266	0.219317	0.109658	−	0.993969i	$-0.465024\pi$
$17$	0.904266	0.219317	0.109658	+	0.993969i	$0.465024\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	−4.31976	−0.942649
$22$	0	0
$23$	−0.548933	−0.114460	−0.0572302	−	0.998361i	$-0.518227\pi$
$23$	−0.548933	−0.114460	−0.0572302	+	0.998361i	$0.518227\pi$
$24$	0	0
$25$	0.324437	0.0648873
$26$	0	0
$27$	−13.8595	−2.66727
$28$	0	0
$29$	−4.32803	−0.803695	−0.401847	−	0.915707i	$-0.631632\pi$
$29$	−4.32803	−0.803695	−0.401847	+	0.915707i	$0.631632\pi$
$30$	0	0
$31$	−1.89392	−0.340159	−0.170079	−	0.985430i	$-0.554402\pi$
$31$	−1.89392	−0.340159	−0.170079	+	0.985430i	$0.554402\pi$
$32$	0	0
$33$	3.21174	0.559092
$34$	0	0
$35$	3.10353	0.524593
$36$	0	0
$37$	−0.546801	−0.0898936	−0.0449468	−	0.998989i	$-0.514312\pi$
$37$	−0.546801	−0.0898936	−0.0449468	+	0.998989i	$0.514312\pi$
$38$	0	0
$39$	4.31976	0.691715
$40$	0	0
$41$	−4.86420	−0.759661	−0.379831	−	0.925056i	$-0.624018\pi$
$41$	−4.86420	−0.759661	−0.379831	+	0.925056i	$0.624018\pi$
$42$	0	0
$43$	−5.24127	−0.799287	−0.399643	−	0.916671i	$-0.630866\pi$
$43$	−5.24127	−0.799287	−0.399643	+	0.916671i	$0.630866\pi$
$44$	0	0
$45$	16.8798	2.51629
$46$	0	0
$47$	5.11346	0.745875	0.372938	−	0.927856i	$-0.378351\pi$
$47$	5.11346	0.745875	0.372938	+	0.927856i	$0.378351\pi$
$48$	0	0
$49$	−5.19100	−0.741571
$50$	0	0
$51$	−2.90427	−0.406679
$52$	0	0
$53$	−3.63971	−0.499952	−0.249976	−	0.968252i	$-0.580423\pi$
$53$	−3.63971	−0.499952	−0.249976	+	0.968252i	$0.580423\pi$
$54$	0	0
$55$	−2.30747	−0.311140
$56$	0	0
$57$	3.21174	0.425405
$58$	0	0
$59$	−13.7786	−1.79382	−0.896912	−	0.442209i	$-0.854195\pi$
$59$	−13.7786	−1.79382	−0.896912	+	0.442209i	$0.854195\pi$
$60$	0	0
$61$	3.90408	0.499866	0.249933	−	0.968263i	$-0.419591\pi$
$61$	3.90408	0.499866	0.249933	+	0.968263i	$0.419591\pi$
$62$	0	0
$63$	9.83897	1.23959
$64$	0	0
$65$	−3.10353	−0.384946
$66$	0	0
$67$	5.72084	0.698912	0.349456	−	0.936953i	$-0.386366\pi$
$67$	5.72084	0.698912	0.349456	+	0.936953i	$0.386366\pi$
$68$	0	0
$69$	1.76303	0.212244
$70$	0	0
$71$	4.49402	0.533342	0.266671	−	0.963788i	$-0.414076\pi$
$71$	4.49402	0.533342	0.266671	+	0.963788i	$0.414076\pi$
$72$	0	0
$73$	1.28228	0.150080	0.0750400	−	0.997181i	$-0.476092\pi$
$73$	1.28228	0.150080	0.0750400	+	0.997181i	$0.476092\pi$
$74$	0	0
$75$	−1.04201	−0.120321
$76$	0	0
$77$	−1.34499	−0.153276
$78$	0	0
$79$	6.80150	0.765228	0.382614	−	0.923908i	$-0.375024\pi$
$79$	6.80150	0.765228	0.382614	+	0.923908i	$0.375024\pi$
$80$	0	0
$81$	22.5674	2.50749
$82$	0	0
$83$	7.06560	0.775551	0.387775	−	0.921754i	$-0.373244\pi$
$83$	7.06560	0.775551	0.387775	+	0.921754i	$0.373244\pi$
$84$	0	0
$85$	2.08657	0.226320
$86$	0	0
$87$	13.9005	1.49029
$88$	0	0
$89$	15.8923	1.68458	0.842289	−	0.539027i	$-0.181208\pi$
$89$	15.8923	1.68458	0.842289	+	0.539027i	$0.181208\pi$
$90$	0	0
$91$	−1.80900	−0.189635
$92$	0	0
$93$	6.08279	0.630756
$94$	0	0
$95$	−2.30747	−0.236742
$96$	0	0
$97$	12.2842	1.24727	0.623635	−	0.781715i	$-0.285655\pi$
$97$	12.2842	1.24727	0.623635	+	0.781715i	$0.285655\pi$
$98$	0	0
$99$	−7.31527	−0.735212
$100$	0	0
$101$	−10.3276	−1.02763	−0.513815	−	0.857901i	$-0.671768\pi$
$101$	−10.3276	−1.02763	−0.513815	+	0.857901i	$0.671768\pi$
$102$	0	0
$103$	−9.65577	−0.951412	−0.475706	−	0.879604i	$-0.657807\pi$
$103$	−9.65577	−0.951412	−0.475706	+	0.879604i	$0.657807\pi$
$104$	0	0
$105$	−9.96774	−0.972752
$106$	0	0
$107$	8.25466	0.798008	0.399004	−	0.916949i	$-0.369356\pi$
$107$	8.25466	0.798008	0.399004	+	0.916949i	$0.369356\pi$
$108$	0	0
$109$	−11.7058	−1.12121	−0.560604	−	0.828084i	$-0.689431\pi$
$109$	−11.7058	−1.12121	−0.560604	+	0.828084i	$0.689431\pi$
$110$	0	0
$111$	1.75618	0.166690
$112$	0	0
$113$	−17.1311	−1.61156	−0.805781	−	0.592213i	$-0.798254\pi$
$113$	−17.1311	−1.61156	−0.805781	+	0.592213i	$0.798254\pi$
$114$	0	0
$115$	−1.26665	−0.118116
$116$	0	0
$117$	−9.83897	−0.909614
$118$	0	0
$119$	1.21623	0.111492
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	15.6226	1.40864
$124$	0	0
$125$	−10.7887	−0.964974
$126$	0	0
$127$	−8.51903	−0.755941	−0.377971	−	0.925818i	$-0.623378\pi$
$127$	−8.51903	−0.755941	−0.377971	+	0.925818i	$0.623378\pi$
$128$	0	0
$129$	16.8336	1.48212
$130$	0	0
$131$	11.0052	0.961530	0.480765	−	0.876849i	$-0.340359\pi$
$131$	11.0052	0.961530	0.480765	+	0.876849i	$0.340359\pi$
$132$	0	0
$133$	−1.34499	−0.116626
$134$	0	0
$135$	−31.9805	−2.75244
$136$	0	0
$137$	−5.62529	−0.480601	−0.240300	−	0.970699i	$-0.577246\pi$
$137$	−5.62529	−0.480601	−0.240300	+	0.970699i	$0.577246\pi$
$138$	0	0
$139$	−11.0613	−0.938211	−0.469105	−	0.883142i	$-0.655424\pi$
$139$	−11.0613	−0.938211	−0.469105	+	0.883142i	$0.655424\pi$
$140$	0	0
$141$	−16.4231	−1.38308
$142$	0	0
$143$	1.34499	0.112474
$144$	0	0
$145$	−9.98681	−0.829360
$146$	0	0
$147$	16.6721	1.37509
$148$	0	0
$149$	−4.56486	−0.373968	−0.186984	−	0.982363i	$-0.559871\pi$
$149$	−4.56486	−0.373968	−0.186984	+	0.982363i	$0.559871\pi$
$150$	0	0
$151$	13.7194	1.11647	0.558236	−	0.829682i	$-0.311478\pi$
$151$	13.7194	1.11647	0.558236	+	0.829682i	$0.311478\pi$
$152$	0	0
$153$	6.61495	0.534787
$154$	0	0
$155$	−4.37018	−0.351021
$156$	0	0
$157$	2.29850	0.183440	0.0917199	−	0.995785i	$-0.470764\pi$
$157$	2.29850	0.183440	0.0917199	+	0.995785i	$0.470764\pi$
$158$	0	0
$159$	11.6898	0.927061
$160$	0	0
$161$	−0.738310	−0.0581870
$162$	0	0
$163$	7.02945	0.550589	0.275294	−	0.961360i	$-0.411225\pi$
$163$	7.02945	0.550589	0.275294	+	0.961360i	$0.411225\pi$
$164$	0	0
$165$	7.41101	0.576946
$166$	0	0
$167$	−23.0788	−1.78589	−0.892946	−	0.450165i	$-0.851365\pi$
$167$	−23.0788	−1.78589	−0.892946	+	0.450165i	$0.851365\pi$
$168$	0	0
$169$	−11.1910	−0.860846
$170$	0	0
$171$	−7.31527	−0.559413
$172$	0	0
$173$	−6.32553	−0.480921	−0.240461	−	0.970659i	$-0.577299\pi$
$173$	−6.32553	−0.480921	−0.240461	+	0.970659i	$0.577299\pi$
$174$	0	0
$175$	0.436365	0.0329861
$176$	0	0
$177$	44.2534	3.32629
$178$	0	0
$179$	−22.6706	−1.69448	−0.847239	−	0.531212i	$-0.821737\pi$
$179$	−22.6706	−1.69448	−0.847239	+	0.531212i	$0.821737\pi$
$180$	0	0
$181$	−19.2524	−1.43102	−0.715509	−	0.698604i	$-0.753805\pi$
$181$	−19.2524	−1.43102	−0.715509	+	0.698604i	$0.753805\pi$
$182$	0	0
$183$	−12.5389	−0.926901
$184$	0	0
$185$	−1.26173	−0.0927642
$186$	0	0
$187$	−0.904266	−0.0661264
$188$	0	0
$189$	−18.6409	−1.35593
$190$	0	0
$191$	−2.30738	−0.166956	−0.0834780	−	0.996510i	$-0.526603\pi$
$191$	−2.30738	−0.166956	−0.0834780	+	0.996510i	$0.526603\pi$
$192$	0	0
$193$	−22.7437	−1.63713	−0.818564	−	0.574415i	$-0.805230\pi$
$193$	−22.7437	−1.63713	−0.818564	+	0.574415i	$0.805230\pi$
$194$	0	0
$195$	9.96774	0.713804
$196$	0	0
$197$	11.4344	0.814668	0.407334	−	0.913279i	$-0.366459\pi$
$197$	11.4344	0.814668	0.407334	+	0.913279i	$0.366459\pi$
$198$	0	0
$199$	−8.50792	−0.603110	−0.301555	−	0.953449i	$-0.597506\pi$
$199$	−8.50792	−0.603110	−0.301555	+	0.953449i	$0.597506\pi$
$200$	0	0
$201$	−18.3738	−1.29599
$202$	0	0
$203$	−5.82116	−0.408565
$204$	0	0
$205$	−11.2240	−0.783920
$206$	0	0
$207$	−4.01560	−0.279103
$208$	0	0
$209$	1.00000	0.0691714
$210$	0	0
$211$	−22.4322	−1.54430	−0.772149	−	0.635441i	$-0.780818\pi$
$211$	−22.4322	−1.54430	−0.772149	+	0.635441i	$0.780818\pi$
$212$	0	0
$213$	−14.4336	−0.988976
$214$	0	0
$215$	−12.0941	−0.824811
$216$	0	0
$217$	−2.54731	−0.172923
$218$	0	0
$219$	−4.11836	−0.278293
$220$	0	0
$221$	−1.21623	−0.0818124
$222$	0	0
$223$	−8.13420	−0.544706	−0.272353	−	0.962197i	$-0.587802\pi$
$223$	−8.13420	−0.544706	−0.272353	+	0.962197i	$0.587802\pi$
$224$	0	0
$225$	2.37334	0.158223
$226$	0	0
$227$	−8.49167	−0.563612	−0.281806	−	0.959471i	$-0.590933\pi$
$227$	−8.49167	−0.563612	−0.281806	+	0.959471i	$0.590933\pi$
$228$	0	0
$229$	−23.5132	−1.55379	−0.776897	−	0.629628i	$-0.783207\pi$
$229$	−23.5132	−1.55379	−0.776897	+	0.629628i	$0.783207\pi$
$230$	0	0
$231$	4.31976	0.284219
$232$	0	0
$233$	−0.186550	−0.0122213	−0.00611064	−	0.999981i	$-0.501945\pi$
$233$	−0.186550	−0.0122213	−0.00611064	+	0.999981i	$0.501945\pi$
$234$	0	0
$235$	11.7992	0.769694
$236$	0	0
$237$	−21.8446	−1.41896
$238$	0	0
$239$	16.0820	1.04026	0.520130	−	0.854087i	$-0.325883\pi$
$239$	16.0820	1.04026	0.520130	+	0.854087i	$0.325883\pi$
$240$	0	0
$241$	29.5369	1.90264	0.951319	−	0.308207i	$-0.0997289\pi$
$241$	29.5369	1.90264	0.951319	+	0.308207i	$0.0997289\pi$
$242$	0	0
$243$	−30.9020	−1.98236
$244$	0	0
$245$	−11.9781	−0.765252
$246$	0	0
$247$	1.34499	0.0855797
$248$	0	0
$249$	−22.6929	−1.43810
$250$	0	0
$251$	8.42537	0.531804	0.265902	−	0.964000i	$-0.414330\pi$
$251$	8.42537	0.531804	0.265902	+	0.964000i	$0.414330\pi$
$252$	0	0
$253$	0.548933	0.0345111
$254$	0	0
$255$	−6.70152	−0.419665
$256$	0	0
$257$	23.5186	1.46705	0.733525	−	0.679662i	$-0.237874\pi$
$257$	23.5186	1.46705	0.733525	+	0.679662i	$0.237874\pi$
$258$	0	0
$259$	−0.735443	−0.0456982
$260$	0	0
$261$	−31.6607	−1.95975
$262$	0	0
$263$	−8.00203	−0.493426	−0.246713	−	0.969089i	$-0.579350\pi$
$263$	−8.00203	−0.493426	−0.246713	+	0.969089i	$0.579350\pi$
$264$	0	0
$265$	−8.39853	−0.515918
$266$	0	0
$267$	−51.0418	−3.12371
$268$	0	0
$269$	−7.63352	−0.465424	−0.232712	−	0.972546i	$-0.574760\pi$
$269$	−7.63352	−0.465424	−0.232712	+	0.972546i	$0.574760\pi$
$270$	0	0
$271$	14.2470	0.865441	0.432720	−	0.901528i	$-0.357554\pi$
$271$	14.2470	0.865441	0.432720	+	0.901528i	$0.357554\pi$
$272$	0	0
$273$	5.81004	0.351640
$274$	0	0
$275$	−0.324437	−0.0195643
$276$	0	0
$277$	23.9558	1.43936	0.719682	−	0.694304i	$-0.244288\pi$
$277$	23.9558	1.43936	0.719682	+	0.694304i	$0.244288\pi$
$278$	0	0
$279$	−13.8546	−0.829451
$280$	0	0
$281$	−3.08649	−0.184125	−0.0920623	−	0.995753i	$-0.529346\pi$
$281$	−3.08649	−0.184125	−0.0920623	+	0.995753i	$0.529346\pi$
$282$	0	0
$283$	6.50272	0.386547	0.193273	−	0.981145i	$-0.438090\pi$
$283$	6.50272	0.386547	0.193273	+	0.981145i	$0.438090\pi$
$284$	0	0
$285$	7.41101	0.438990
$286$	0	0
$287$	−6.54231	−0.386180
$288$	0	0
$289$	−16.1823	−0.951900
$290$	0	0
$291$	−39.4536	−2.31281
$292$	0	0
$293$	5.43591	0.317569	0.158785	−	0.987313i	$-0.449242\pi$
$293$	5.43591	0.317569	0.158785	+	0.987313i	$0.449242\pi$
$294$	0	0
$295$	−31.7938	−1.85111
$296$	0	0
$297$	13.8595	0.804211
$298$	0	0
$299$	0.738310	0.0426976
$300$	0	0
$301$	−7.04946	−0.406324
$302$	0	0
$303$	33.1694	1.90553
$304$	0	0
$305$	9.00856	0.515829
$306$	0	0
$307$	−9.01422	−0.514469	−0.257234	−	0.966349i	$-0.582811\pi$
$307$	−9.01422	−0.514469	−0.257234	+	0.966349i	$0.582811\pi$
$308$	0	0
$309$	31.0118	1.76420
$310$	0	0
$311$	−28.2244	−1.60046	−0.800230	−	0.599693i	$-0.795289\pi$
$311$	−28.2244	−1.60046	−0.800230	+	0.599693i	$0.795289\pi$
$312$	0	0
$313$	−3.14170	−0.177580	−0.0887898	−	0.996050i	$-0.528300\pi$
$313$	−3.14170	−0.177580	−0.0887898	+	0.996050i	$0.528300\pi$
$314$	0	0
$315$	22.7032	1.27918
$316$	0	0
$317$	18.9722	1.06559	0.532793	−	0.846246i	$-0.321143\pi$
$317$	18.9722	1.06559	0.532793	+	0.846246i	$0.321143\pi$
$318$	0	0
$319$	4.32803	0.242323
$320$	0	0
$321$	−26.5118	−1.47974
$322$	0	0
$323$	−0.904266	−0.0503147
$324$	0	0
$325$	−0.436365	−0.0242051
$326$	0	0
$327$	37.5959	2.07906
$328$	0	0
$329$	6.87756	0.379172
$330$	0	0
$331$	3.88149	0.213346	0.106673	−	0.994294i	$-0.465980\pi$
$331$	3.88149	0.213346	0.106673	+	0.994294i	$0.465980\pi$
$332$	0	0
$333$	−4.00000	−0.219199
$334$	0	0
$335$	13.2007	0.721230
$336$	0	0
$337$	24.6995	1.34547	0.672734	−	0.739885i	$-0.265120\pi$
$337$	24.6995	1.34547	0.672734	+	0.739885i	$0.265120\pi$
$338$	0	0
$339$	55.0208	2.98832
$340$	0	0
$341$	1.89392	0.102562
$342$	0	0
$343$	−16.3968	−0.885343
$344$	0	0
$345$	4.06815	0.219022
$346$	0	0
$347$	33.7555	1.81209	0.906046	−	0.423179i	$-0.139086\pi$
$347$	33.7555	1.81209	0.906046	+	0.423179i	$0.139086\pi$
$348$	0	0
$349$	−33.1420	−1.77405	−0.887026	−	0.461720i	$-0.847232\pi$
$349$	−33.1420	−1.77405	−0.887026	+	0.461720i	$0.847232\pi$
$350$	0	0
$351$	18.6409	0.994980
$352$	0	0
$353$	14.0448	0.747527	0.373764	−	0.927524i	$-0.378067\pi$
$353$	14.0448	0.747527	0.373764	+	0.927524i	$0.378067\pi$
$354$	0	0
$355$	10.3698	0.550374
$356$	0	0
$357$	−3.90621	−0.206739
$358$	0	0
$359$	−35.9721	−1.89853	−0.949266	−	0.314473i	$-0.898172\pi$
$359$	−35.9721	−1.89853	−0.949266	+	0.314473i	$0.898172\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	−3.21174	−0.168573
$364$	0	0
$365$	2.95884	0.154873
$366$	0	0
$367$	33.4752	1.74739	0.873697	−	0.486471i	$-0.161716\pi$
$367$	33.4752	1.74739	0.873697	+	0.486471i	$0.161716\pi$
$368$	0	0
$369$	−35.5830	−1.85238
$370$	0	0
$371$	−4.89538	−0.254155
$372$	0	0
$373$	−18.2471	−0.944801	−0.472400	−	0.881384i	$-0.656612\pi$
$373$	−18.2471	−0.944801	−0.472400	+	0.881384i	$0.656612\pi$
$374$	0	0
$375$	34.6506	1.78935
$376$	0	0
$377$	5.82116	0.299805
$378$	0	0
$379$	11.7401	0.603050	0.301525	−	0.953458i	$-0.402504\pi$
$379$	11.7401	0.603050	0.301525	+	0.953458i	$0.402504\pi$
$380$	0	0
$381$	27.3609	1.40174
$382$	0	0
$383$	−2.42259	−0.123788	−0.0618942	−	0.998083i	$-0.519714\pi$
$383$	−2.42259	−0.123788	−0.0618942	+	0.998083i	$0.519714\pi$
$384$	0	0
$385$	−3.10353	−0.158171
$386$	0	0
$387$	−38.3413	−1.94900
$388$	0	0
$389$	−20.7267	−1.05088	−0.525442	−	0.850829i	$-0.676100\pi$
$389$	−20.7267	−1.05088	−0.525442	+	0.850829i	$0.676100\pi$
$390$	0	0
$391$	−0.496381	−0.0251031
$392$	0	0
$393$	−35.3459	−1.78296
$394$	0	0
$395$	15.6943	0.789665
$396$	0	0
$397$	−5.89658	−0.295941	−0.147971	−	0.988992i	$-0.547274\pi$
$397$	−5.89658	−0.295941	−0.147971	+	0.988992i	$0.547274\pi$
$398$	0	0
$399$	4.31976	0.216259
$400$	0	0
$401$	−31.9214	−1.59408	−0.797040	−	0.603927i	$-0.793602\pi$
$401$	−31.9214	−1.59408	−0.797040	+	0.603927i	$0.793602\pi$
$402$	0	0
$403$	2.54731	0.126891
$404$	0	0
$405$	52.0737	2.58756
$406$	0	0
$407$	0.546801	0.0271039
$408$	0	0
$409$	17.9584	0.887985	0.443993	−	0.896030i	$-0.353562\pi$
$409$	17.9584	0.887985	0.443993	+	0.896030i	$0.353562\pi$
$410$	0	0
$411$	18.0670	0.891178
$412$	0	0
$413$	−18.5321	−0.911906
$414$	0	0
$415$	16.3037	0.800317
$416$	0	0
$417$	35.5261	1.73972
$418$	0	0
$419$	1.88319	0.0920000	0.0460000	−	0.998941i	$-0.485353\pi$
$419$	1.88319	0.0920000	0.0460000	+	0.998941i	$0.485353\pi$
$420$	0	0
$421$	−28.1722	−1.37303	−0.686514	−	0.727117i	$-0.740860\pi$
$421$	−28.1722	−1.37303	−0.686514	+	0.727117i	$0.740860\pi$
$422$	0	0
$423$	37.4064	1.81876
$424$	0	0
$425$	0.293377	0.0142309
$426$	0	0
$427$	5.25095	0.254111
$428$	0	0
$429$	−4.31976	−0.208560
$430$	0	0
$431$	−2.23684	−0.107745	−0.0538724	−	0.998548i	$-0.517156\pi$
$431$	−2.23684	−0.107745	−0.0538724	+	0.998548i	$0.517156\pi$
$432$	0	0
$433$	6.74878	0.324326	0.162163	−	0.986764i	$-0.448153\pi$
$433$	6.74878	0.324326	0.162163	+	0.986764i	$0.448153\pi$
$434$	0	0
$435$	32.0750	1.53788
$436$	0	0
$437$	0.548933	0.0262590
$438$	0	0
$439$	37.0940	1.77040	0.885201	−	0.465210i	$-0.154021\pi$
$439$	37.0940	1.77040	0.885201	+	0.465210i	$0.154021\pi$
$440$	0	0
$441$	−37.9736	−1.80827
$442$	0	0
$443$	13.9416	0.662387	0.331193	−	0.943563i	$-0.392549\pi$
$443$	13.9416	0.662387	0.331193	+	0.943563i	$0.392549\pi$
$444$	0	0
$445$	36.6710	1.73837
$446$	0	0
$447$	14.6611	0.693448
$448$	0	0
$449$	−10.4412	−0.492753	−0.246376	−	0.969174i	$-0.579240\pi$
$449$	−10.4412	−0.492753	−0.246376	+	0.969174i	$0.579240\pi$
$450$	0	0
$451$	4.86420	0.229046
$452$	0	0
$453$	−44.0632	−2.07027
$454$	0	0
$455$	−4.17422	−0.195691
$456$	0	0
$457$	−18.8859	−0.883443	−0.441722	−	0.897152i	$-0.645632\pi$
$457$	−18.8859	−0.883443	−0.441722	+	0.897152i	$0.645632\pi$
$458$	0	0
$459$	−12.5327	−0.584976
$460$	0	0
$461$	35.0739	1.63355	0.816777	−	0.576954i	$-0.195759\pi$
$461$	35.0739	1.63355	0.816777	+	0.576954i	$0.195759\pi$
$462$	0	0
$463$	−4.84941	−0.225371	−0.112686	−	0.993631i	$-0.535945\pi$
$463$	−4.84941	−0.225371	−0.112686	+	0.993631i	$0.535945\pi$
$464$	0	0
$465$	14.0359	0.650899
$466$	0	0
$467$	0.400793	0.0185465	0.00927325	−	0.999957i	$-0.497048\pi$
$467$	0.400793	0.0185465	0.00927325	+	0.999957i	$0.497048\pi$
$468$	0	0
$469$	7.69448	0.355298
$470$	0	0
$471$	−7.38217	−0.340152
$472$	0	0
$473$	5.24127	0.240994
$474$	0	0
$475$	−0.324437	−0.0148862
$476$	0	0
$477$	−26.6255	−1.21910
$478$	0	0
$479$	−19.9633	−0.912147	−0.456074	−	0.889942i	$-0.650745\pi$
$479$	−19.9633	−0.912147	−0.456074	+	0.889942i	$0.650745\pi$
$480$	0	0
$481$	0.735443	0.0335333
$482$	0	0
$483$	2.37126	0.107896
$484$	0	0
$485$	28.3454	1.28710
$486$	0	0
$487$	−20.9214	−0.948039	−0.474019	−	0.880514i	$-0.657197\pi$
$487$	−20.9214	−0.948039	−0.474019	+	0.880514i	$0.657197\pi$
$488$	0	0
$489$	−22.5768	−1.02096
$490$	0	0
$491$	−22.6176	−1.02072	−0.510358	−	0.859962i	$-0.670487\pi$
$491$	−22.6176	−1.02072	−0.510358	+	0.859962i	$0.670487\pi$
$492$	0	0
$493$	−3.91369	−0.176264
$494$	0	0
$495$	−16.8798	−0.758691
$496$	0	0
$497$	6.04442	0.271129
$498$	0	0
$499$	−25.4367	−1.13870	−0.569351	−	0.822095i	$-0.692805\pi$
$499$	−25.4367	−1.13870	−0.569351	+	0.822095i	$0.692805\pi$
$500$	0	0
$501$	74.1231	3.31158
$502$	0	0
$503$	−0.0412059	−0.00183728	−0.000918639	−	1.00000i	$-0.500292\pi$
$503$	−0.0412059	−0.00183728	−0.000918639		1.00000i	$0.500292\pi$
$504$	0	0
$505$	−23.8306	−1.06045
$506$	0	0
$507$	35.9426	1.59627
$508$	0	0
$509$	22.7238	1.00722	0.503608	−	0.863932i	$-0.332006\pi$
$509$	22.7238	1.00722	0.503608	+	0.863932i	$0.332006\pi$
$510$	0	0
$511$	1.72466	0.0762945
$512$	0	0
$513$	13.8595	0.611913
$514$	0	0
$515$	−22.2804	−0.981794
$516$	0	0
$517$	−5.11346	−0.224890
$518$	0	0
$519$	20.3160	0.891772
$520$	0	0
$521$	−42.8088	−1.87549	−0.937744	−	0.347327i	$-0.887089\pi$
$521$	−42.8088	−1.87549	−0.937744	+	0.347327i	$0.887089\pi$
$522$	0	0
$523$	12.3759	0.541160	0.270580	−	0.962698i	$-0.412785\pi$
$523$	12.3759	0.541160	0.270580	+	0.962698i	$0.412785\pi$
$524$	0	0
$525$	−1.40149	−0.0611660
$526$	0	0
$527$	−1.71261	−0.0746025
$528$	0	0
$529$	−22.6987	−0.986899
$530$	0	0
$531$	−100.794	−4.37410
$532$	0	0
$533$	6.54231	0.283379
$534$	0	0
$535$	19.0474	0.823491
$536$	0	0
$537$	72.8119	3.14207
$538$	0	0
$539$	5.19100	0.223592
$540$	0	0
$541$	−8.66550	−0.372559	−0.186279	−	0.982497i	$-0.559643\pi$
$541$	−8.66550	−0.372559	−0.186279	+	0.982497i	$0.559643\pi$
$542$	0	0
$543$	61.8336	2.65353
$544$	0	0
$545$	−27.0107	−1.15701
$546$	0	0
$547$	−18.1983	−0.778103	−0.389051	−	0.921216i	$-0.627197\pi$
$547$	−18.1983	−0.778103	−0.389051	+	0.921216i	$0.627197\pi$
$548$	0	0
$549$	28.5594	1.21889
$550$	0	0
$551$	4.32803	0.184380
$552$	0	0
$553$	9.14795	0.389011
$554$	0	0
$555$	4.05235	0.172013
$556$	0	0
$557$	31.5397	1.33638	0.668190	−	0.743991i	$-0.267069\pi$
$557$	31.5397	1.33638	0.668190	+	0.743991i	$0.267069\pi$
$558$	0	0
$559$	7.04946	0.298161
$560$	0	0
$561$	2.90427	0.122618
$562$	0	0
$563$	−23.2066	−0.978044	−0.489022	−	0.872272i	$-0.662646\pi$
$563$	−23.2066	−0.978044	−0.489022	+	0.872272i	$0.662646\pi$
$564$	0	0
$565$	−39.5297	−1.66303
$566$	0	0
$567$	30.3529	1.27470
$568$	0	0
$569$	14.0539	0.589171	0.294586	−	0.955625i	$-0.404818\pi$
$569$	14.0539	0.589171	0.294586	+	0.955625i	$0.404818\pi$
$570$	0	0
$571$	−42.5174	−1.77930	−0.889648	−	0.456646i	$-0.849051\pi$
$571$	−42.5174	−1.77930	−0.889648	+	0.456646i	$0.849051\pi$
$572$	0	0
$573$	7.41069	0.309586
$574$	0	0
$575$	−0.178094	−0.00742704
$576$	0	0
$577$	−12.9888	−0.540732	−0.270366	−	0.962758i	$-0.587145\pi$
$577$	−12.9888	−0.540732	−0.270366	+	0.962758i	$0.587145\pi$
$578$	0	0
$579$	73.0469	3.03573
$580$	0	0
$581$	9.50317	0.394258
$582$	0	0
$583$	3.63971	0.150741
$584$	0	0
$585$	−22.7032	−0.938661
$586$	0	0
$587$	25.4076	1.04868	0.524341	−	0.851508i	$-0.324312\pi$
$587$	25.4076	1.04868	0.524341	+	0.851508i	$0.324312\pi$
$588$	0	0
$589$	1.89392	0.0780378
$590$	0	0
$591$	−36.7243	−1.51064
$592$	0	0
$593$	11.3346	0.465456	0.232728	−	0.972542i	$-0.425235\pi$
$593$	11.3346	0.465456	0.232728	+	0.972542i	$0.425235\pi$
$594$	0	0
$595$	2.80642	0.115052
$596$	0	0
$597$	27.3252	1.11835
$598$	0	0
$599$	44.3934	1.81387	0.906933	−	0.421276i	$-0.138418\pi$
$599$	44.3934	1.81387	0.906933	+	0.421276i	$0.138418\pi$
$600$	0	0
$601$	−23.5195	−0.959380	−0.479690	−	0.877438i	$-0.659251\pi$
$601$	−23.5195	−0.959380	−0.479690	+	0.877438i	$0.659251\pi$
$602$	0	0
$603$	41.8495	1.70424
$604$	0	0
$605$	2.30747	0.0938122
$606$	0	0
$607$	33.5476	1.36165	0.680827	−	0.732444i	$-0.261621\pi$
$607$	33.5476	1.36165	0.680827	+	0.732444i	$0.261621\pi$
$608$	0	0
$609$	18.6960	0.757602
$610$	0	0
$611$	−6.87756	−0.278236
$612$	0	0
$613$	−25.6571	−1.03628	−0.518141	−	0.855295i	$-0.673376\pi$
$613$	−25.6571	−1.03628	−0.518141	+	0.855295i	$0.673376\pi$
$614$	0	0
$615$	36.0487	1.45362
$616$	0	0
$617$	7.96480	0.320651	0.160325	−	0.987064i	$-0.448746\pi$
$617$	7.96480	0.320651	0.160325	+	0.987064i	$0.448746\pi$
$618$	0	0
$619$	−24.9617	−1.00329	−0.501647	−	0.865072i	$-0.667272\pi$
$619$	−24.9617	−1.00329	−0.501647	+	0.865072i	$0.667272\pi$
$620$	0	0
$621$	7.60795	0.305297
$622$	0	0
$623$	21.3750	0.856370
$624$	0	0
$625$	−26.5169	−1.06068
$626$	0	0
$627$	−3.21174	−0.128265
$628$	0	0
$629$	−0.494454	−0.0197152
$630$	0	0
$631$	−13.8618	−0.551831	−0.275916	−	0.961182i	$-0.588981\pi$
$631$	−13.8618	−0.551831	−0.275916	+	0.961182i	$0.588981\pi$
$632$	0	0
$633$	72.0465	2.86359
$634$	0	0
$635$	−19.6574	−0.780082
$636$	0	0
$637$	6.98185	0.276631
$638$	0	0
$639$	32.8750	1.30052
$640$	0	0
$641$	38.4522	1.51877	0.759386	−	0.650640i	$-0.225499\pi$
$641$	38.4522	1.51877	0.759386	+	0.650640i	$0.225499\pi$
$642$	0	0
$643$	32.7622	1.29202	0.646008	−	0.763331i	$-0.276437\pi$
$643$	32.7622	1.29202	0.646008	+	0.763331i	$0.276437\pi$
$644$	0	0
$645$	38.8431	1.52945
$646$	0	0
$647$	−30.8413	−1.21250	−0.606248	−	0.795276i	$-0.707326\pi$
$647$	−30.8413	−1.21250	−0.606248	+	0.795276i	$0.707326\pi$
$648$	0	0
$649$	13.7786	0.540858
$650$	0	0
$651$	8.18130	0.320650
$652$	0	0
$653$	−7.13589	−0.279249	−0.139624	−	0.990205i	$-0.544590\pi$
$653$	−7.13589	−0.279249	−0.139624	+	0.990205i	$0.544590\pi$
$654$	0	0
$655$	25.3942	0.992235
$656$	0	0
$657$	9.38026	0.365959
$658$	0	0
$659$	4.67399	0.182073	0.0910364	−	0.995848i	$-0.470982\pi$
$659$	4.67399	0.182073	0.0910364	+	0.995848i	$0.470982\pi$
$660$	0	0
$661$	31.3405	1.21900	0.609502	−	0.792784i	$-0.291369\pi$
$661$	31.3405	1.21900	0.609502	+	0.792784i	$0.291369\pi$
$662$	0	0
$663$	3.90621	0.151705
$664$	0	0
$665$	−3.10353	−0.120350
$666$	0	0
$667$	2.37580	0.0919913
$668$	0	0
$669$	26.1249	1.01005
$670$	0	0
$671$	−3.90408	−0.150715
$672$	0	0
$673$	−18.1600	−0.700018	−0.350009	−	0.936746i	$-0.613821\pi$
$673$	−18.1600	−0.700018	−0.350009	+	0.936746i	$0.613821\pi$
$674$	0	0
$675$	−4.49654	−0.173072
$676$	0	0
$677$	−32.0875	−1.23322	−0.616611	−	0.787268i	$-0.711495\pi$
$677$	−32.0875	−1.23322	−0.616611	+	0.787268i	$0.711495\pi$
$678$	0	0
$679$	16.5221	0.634061
$680$	0	0
$681$	27.2730	1.04510
$682$	0	0
$683$	−4.30997	−0.164917	−0.0824583	−	0.996595i	$-0.526277\pi$
$683$	−4.30997	−0.164917	−0.0824583	+	0.996595i	$0.526277\pi$
$684$	0	0
$685$	−12.9802	−0.495948
$686$	0	0
$687$	75.5182	2.88120
$688$	0	0
$689$	4.89538	0.186499
$690$	0	0
$691$	−36.2358	−1.37848	−0.689238	−	0.724535i	$-0.742055\pi$
$691$	−36.2358	−1.37848	−0.689238	+	0.724535i	$0.742055\pi$
$692$	0	0
$693$	−9.83897	−0.373752
$694$	0	0
$695$	−25.5238	−0.968171
$696$	0	0
$697$	−4.39853	−0.166606
$698$	0	0
$699$	0.599149	0.0226619
$700$	0	0
$701$	6.19008	0.233796	0.116898	−	0.993144i	$-0.462705\pi$
$701$	6.19008	0.233796	0.116898	+	0.993144i	$0.462705\pi$
$702$	0	0
$703$	0.546801	0.0206230
$704$	0	0
$705$	−37.8959	−1.42724
$706$	0	0
$707$	−13.8905	−0.522405
$708$	0	0
$709$	30.6859	1.15243	0.576216	−	0.817297i	$-0.304529\pi$
$709$	30.6859	1.15243	0.576216	+	0.817297i	$0.304529\pi$
$710$	0	0
$711$	49.7548	1.86595
$712$	0	0
$713$	1.03964	0.0389347
$714$	0	0
$715$	3.10353	0.116066
$716$	0	0
$717$	−51.6513	−1.92895
$718$	0	0
$719$	21.6704	0.808171	0.404085	−	0.914721i	$-0.367590\pi$
$719$	21.6704	0.808171	0.404085	+	0.914721i	$0.367590\pi$
$720$	0	0
$721$	−12.9869	−0.483658
$722$	0	0
$723$	−94.8648	−3.52806
$724$	0	0
$725$	−1.40417	−0.0521496
$726$	0	0
$727$	1.88637	0.0699614	0.0349807	−	0.999388i	$-0.488863\pi$
$727$	1.88637	0.0699614	0.0349807	+	0.999388i	$0.488863\pi$
$728$	0	0
$729$	31.5469	1.16841
$730$	0	0
$731$	−4.73950	−0.175297
$732$	0	0
$733$	−10.7015	−0.395270	−0.197635	−	0.980276i	$-0.563326\pi$
$733$	−10.7015	−0.395270	−0.197635	+	0.980276i	$0.563326\pi$
$734$	0	0
$735$	38.4705	1.41901
$736$	0	0
$737$	−5.72084	−0.210730
$738$	0	0
$739$	−11.6797	−0.429644	−0.214822	−	0.976653i	$-0.568917\pi$
$739$	−11.6797	−0.429644	−0.214822	+	0.976653i	$0.568917\pi$
$740$	0	0
$741$	−4.31976	−0.158690
$742$	0	0
$743$	−27.2152	−0.998428	−0.499214	−	0.866479i	$-0.666378\pi$
$743$	−27.2152	−0.998428	−0.499214	+	0.866479i	$0.666378\pi$
$744$	0	0
$745$	−10.5333	−0.385910
$746$	0	0
$747$	51.6868	1.89112
$748$	0	0
$749$	11.1024	0.405674
$750$	0	0
$751$	4.26615	0.155674	0.0778371	−	0.996966i	$-0.475199\pi$
$751$	4.26615	0.155674	0.0778371	+	0.996966i	$0.475199\pi$
$752$	0	0
$753$	−27.0601	−0.986124
$754$	0	0
$755$	31.6572	1.15212
$756$	0	0
$757$	−14.2953	−0.519571	−0.259786	−	0.965666i	$-0.583652\pi$
$757$	−14.2953	−0.519571	−0.259786	+	0.965666i	$0.583652\pi$
$758$	0	0
$759$	−1.76303	−0.0639939
$760$	0	0
$761$	−4.12180	−0.149415	−0.0747076	−	0.997205i	$-0.523802\pi$
$761$	−4.12180	−0.149415	−0.0747076	+	0.997205i	$0.523802\pi$
$762$	0	0
$763$	−15.7441	−0.569976
$764$	0	0
$765$	15.2638	0.551865
$766$	0	0
$767$	18.5321	0.669156
$768$	0	0
$769$	24.0195	0.866164	0.433082	−	0.901355i	$-0.357426\pi$
$769$	24.0195	0.866164	0.433082	+	0.901355i	$0.357426\pi$
$770$	0	0
$771$	−75.5357	−2.72035
$772$	0	0
$773$	26.4504	0.951354	0.475677	−	0.879620i	$-0.342203\pi$
$773$	26.4504	0.951354	0.475677	+	0.879620i	$0.342203\pi$
$774$	0	0
$775$	−0.614459	−0.0220720
$776$	0	0
$777$	2.36205	0.0847381
$778$	0	0
$779$	4.86420	0.174278
$780$	0	0
$781$	−4.49402	−0.160809
$782$	0	0
$783$	59.9844	2.14367
$784$	0	0
$785$	5.30372	0.189298
$786$	0	0
$787$	−11.0855	−0.395157	−0.197578	−	0.980287i	$-0.563308\pi$
$787$	−11.0855	−0.395157	−0.197578	+	0.980287i	$0.563308\pi$
$788$	0	0
$789$	25.7004	0.914959
$790$	0	0
$791$	−23.0412	−0.819252
$792$	0	0
$793$	−5.25095	−0.186467
$794$	0	0
$795$	26.9739	0.956666
$796$	0	0
$797$	48.0917	1.70349	0.851747	−	0.523953i	$-0.175543\pi$
$797$	48.0917	1.70349	0.851747	+	0.523953i	$0.175543\pi$
$798$	0	0
$799$	4.62393	0.163583
$800$	0	0
$801$	116.256	4.10771
$802$	0	0
$803$	−1.28228	−0.0452508
$804$	0	0
$805$	−1.70363	−0.0600451
$806$	0	0
$807$	24.5169	0.863035
$808$	0	0
$809$	42.8287	1.50578	0.752888	−	0.658148i	$-0.228660\pi$
$809$	42.8287	1.50578	0.752888	+	0.658148i	$0.228660\pi$
$810$	0	0
$811$	−30.4487	−1.06920	−0.534599	−	0.845106i	$-0.679537\pi$
$811$	−30.4487	−1.06920	−0.534599	+	0.845106i	$0.679537\pi$
$812$	0	0
$813$	−45.7575	−1.60479
$814$	0	0
$815$	16.2203	0.568171
$816$	0	0
$817$	5.24127	0.183369
$818$	0	0
$819$	−13.2333	−0.462410
$820$	0	0
$821$	29.4900	1.02921	0.514604	−	0.857428i	$-0.327939\pi$
$821$	29.4900	1.02921	0.514604	+	0.857428i	$0.327939\pi$
$822$	0	0
$823$	12.9097	0.450003	0.225002	−	0.974358i	$-0.427761\pi$
$823$	12.9097	0.450003	0.225002	+	0.974358i	$0.427761\pi$
$824$	0	0
$825$	1.04201	0.0362780
$826$	0	0
$827$	−51.5466	−1.79245	−0.896225	−	0.443600i	$-0.853701\pi$
$827$	−51.5466	−1.79245	−0.896225	+	0.443600i	$0.853701\pi$
$828$	0	0
$829$	6.89217	0.239375	0.119687	−	0.992812i	$-0.461811\pi$
$829$	6.89217	0.239375	0.119687	+	0.992812i	$0.461811\pi$
$830$	0	0
$831$	−76.9397	−2.66901
$832$	0	0
$833$	−4.69404	−0.162639
$834$	0	0
$835$	−53.2537	−1.84292
$836$	0	0
$837$	26.2489	0.907295
$838$	0	0
$839$	23.2397	0.802323	0.401162	−	0.916007i	$-0.368607\pi$
$839$	23.2397	0.802323	0.401162	+	0.916007i	$0.368607\pi$
$840$	0	0
$841$	−10.2682	−0.354075
$842$	0	0
$843$	9.91300	0.341422
$844$	0	0
$845$	−25.8229	−0.888336
$846$	0	0
$847$	1.34499	0.0462144
$848$	0	0
$849$	−20.8850	−0.716773
$850$	0	0
$851$	0.300157	0.0102893
$852$	0	0
$853$	39.0161	1.33589	0.667943	−	0.744212i	$-0.267175\pi$
$853$	39.0161	1.33589	0.667943	+	0.744212i	$0.267175\pi$
$854$	0	0
$855$	−16.8798	−0.577277
$856$	0	0
$857$	36.2608	1.23865	0.619323	−	0.785137i	$-0.287407\pi$
$857$	36.2608	1.23865	0.619323	+	0.785137i	$0.287407\pi$
$858$	0	0
$859$	−14.1249	−0.481937	−0.240968	−	0.970533i	$-0.577465\pi$
$859$	−14.1249	−0.481937	−0.240968	+	0.970533i	$0.577465\pi$
$860$	0	0
$861$	21.0122	0.716094
$862$	0	0
$863$	−54.5442	−1.85671	−0.928353	−	0.371699i	$-0.878775\pi$
$863$	−54.5442	−1.85671	−0.928353	+	0.371699i	$0.878775\pi$
$864$	0	0
$865$	−14.5960	−0.496279
$866$	0	0
$867$	51.9733	1.76511
$868$	0	0
$869$	−6.80150	−0.230725
$870$	0	0
$871$	−7.69448	−0.260717
$872$	0	0
$873$	89.8622	3.04137
$874$	0	0
$875$	−14.5108	−0.490553
$876$	0	0
$877$	5.59673	0.188988	0.0944941	−	0.995525i	$-0.469877\pi$
$877$	5.59673	0.188988	0.0944941	+	0.995525i	$0.469877\pi$
$878$	0	0
$879$	−17.4587	−0.588869
$880$	0	0
$881$	11.9259	0.401793	0.200897	−	0.979612i	$-0.435614\pi$
$881$	11.9259	0.401793	0.200897	+	0.979612i	$0.435614\pi$
$882$	0	0
$883$	32.3029	1.08708	0.543540	−	0.839383i	$-0.317084\pi$
$883$	32.3029	1.08708	0.543540	+	0.839383i	$0.317084\pi$
$884$	0	0
$885$	102.113	3.43251
$886$	0	0
$887$	−44.1671	−1.48299	−0.741494	−	0.670960i	$-0.765882\pi$
$887$	−44.1671	−1.48299	−0.741494	+	0.670960i	$0.765882\pi$
$888$	0	0
$889$	−11.4580	−0.384290
$890$	0	0
$891$	−22.5674	−0.756036
$892$	0	0
$893$	−5.11346	−0.171115
$894$	0	0
$895$	−52.3117	−1.74859
$896$	0	0
$897$	−2.37126	−0.0791741
$898$	0	0
$899$	8.19696	0.273384
$900$	0	0
$901$	−3.29126	−0.109648
$902$	0	0
$903$	22.6410	0.753447
$904$	0	0
$905$	−44.4244	−1.47672
$906$	0	0
$907$	−5.21952	−0.173311	−0.0866557	−	0.996238i	$-0.527618\pi$
$907$	−5.21952	−0.173311	−0.0866557	+	0.996238i	$0.527618\pi$
$908$	0	0
$909$	−75.5489	−2.50580
$910$	0	0
$911$	51.0254	1.69055	0.845274	−	0.534333i	$-0.179437\pi$
$911$	51.0254	1.69055	0.845274	+	0.534333i	$0.179437\pi$
$912$	0	0
$913$	−7.06560	−0.233837
$914$	0	0
$915$	−28.9332	−0.956501
$916$	0	0
$917$	14.8019	0.488802
$918$	0	0
$919$	−36.0823	−1.19024	−0.595122	−	0.803635i	$-0.702896\pi$
$919$	−36.0823	−1.19024	−0.595122	+	0.803635i	$0.702896\pi$
$920$	0	0
$921$	28.9513	0.953978
$922$	0	0
$923$	−6.04442	−0.198955
$924$	0	0
$925$	−0.177402	−0.00583296
$926$	0	0
$927$	−70.6346	−2.31994
$928$	0	0
$929$	−18.3679	−0.602632	−0.301316	−	0.953524i	$-0.597426\pi$
$929$	−18.3679	−0.602632	−0.301316	+	0.953524i	$0.597426\pi$
$930$	0	0
$931$	5.19100	0.170128
$932$	0	0
$933$	90.6495	2.96773
$934$	0	0
$935$	−2.08657	−0.0682381
$936$	0	0
$937$	30.9327	1.01053	0.505264	−	0.862965i	$-0.331395\pi$
$937$	30.9327	1.01053	0.505264	+	0.862965i	$0.331395\pi$
$938$	0	0
$939$	10.0903	0.329286
$940$	0	0
$941$	33.1792	1.08161	0.540806	−	0.841148i	$-0.318119\pi$
$941$	33.1792	1.08161	0.540806	+	0.841148i	$0.318119\pi$
$942$	0	0
$943$	2.67012	0.0869512
$944$	0	0
$945$	−43.0135	−1.39923
$946$	0	0
$947$	−21.9471	−0.713184	−0.356592	−	0.934260i	$-0.616061\pi$
$947$	−21.9471	−0.713184	−0.356592	+	0.934260i	$0.616061\pi$
$948$	0	0
$949$	−1.72466	−0.0559849
$950$	0	0
$951$	−60.9338	−1.97591
$952$	0	0
$953$	41.4001	1.34108	0.670540	−	0.741873i	$-0.266062\pi$
$953$	41.4001	1.34108	0.670540	+	0.741873i	$0.266062\pi$
$954$	0	0
$955$	−5.32421	−0.172287
$956$	0	0
$957$	−13.9005	−0.449339
$958$	0	0
$959$	−7.56596	−0.244318
$960$	0	0
$961$	−27.4131	−0.884292
$962$	0	0
$963$	60.3851	1.94588
$964$	0	0
$965$	−52.4805	−1.68941
$966$	0	0
$967$	16.5235	0.531358	0.265679	−	0.964062i	$-0.414404\pi$
$967$	16.5235	0.531358	0.265679	+	0.964062i	$0.414404\pi$
$968$	0	0
$969$	2.90427	0.0932985
$970$	0	0
$971$	14.0515	0.450933	0.225467	−	0.974251i	$-0.427609\pi$
$971$	14.0515	0.450933	0.225467	+	0.974251i	$0.427609\pi$
$972$	0	0
$973$	−14.8774	−0.476948
$974$	0	0
$975$	1.40149	0.0448836
$976$	0	0
$977$	7.28151	0.232956	0.116478	−	0.993193i	$-0.462840\pi$
$977$	7.28151	0.232956	0.116478	+	0.993193i	$0.462840\pi$
$978$	0	0
$979$	−15.8923	−0.507919
$980$	0	0
$981$	−85.6308	−2.73398
$982$	0	0
$983$	−61.8115	−1.97148	−0.985740	−	0.168277i	$-0.946180\pi$
$983$	−61.8115	−1.97148	−0.985740	+	0.168277i	$0.946180\pi$
$984$	0	0
$985$	26.3846	0.840683
$986$	0	0
$987$	−22.0889	−0.703098
$988$	0	0
$989$	2.87711	0.0914867
$990$	0	0
$991$	34.0846	1.08273	0.541367	−	0.840786i	$-0.317907\pi$
$991$	34.0846	1.08273	0.541367	+	0.840786i	$0.317907\pi$
$992$	0	0
$993$	−12.4663	−0.395607
$994$	0	0
$995$	−19.6318	−0.622370
$996$	0	0
$997$	−17.6127	−0.557800	−0.278900	−	0.960320i	$-0.589970\pi$
$997$	−17.6127	−0.557800	−0.278900	+	0.960320i	$0.589970\pi$
$998$	0	0
$999$	7.57841	0.239770

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3344.2.a.v.1.1		6
4.3	odd	2		1672.2.a.j.1.6	✓	6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1672.2.a.j.1.6	✓	6	4.3	odd	2
3344.2.a.v.1.1		6	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 3344 = 2^{4} \cdot 11 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3344.a (trivial)

\(f(q)\)	\(=\)	\(q-3.21174 q^{3} +2.30747 q^{5} +1.34499 q^{7} +7.31527 q^{9} -1.00000 q^{11} -1.34499 q^{13} -7.41101 q^{15} +0.904266 q^{17} -1.00000 q^{19} -4.31976 q^{21} -0.548933 q^{23} +0.324437 q^{25} -13.8595 q^{27} -4.32803 q^{29} -1.89392 q^{31} +3.21174 q^{33} +3.10353 q^{35} -0.546801 q^{37} +4.31976 q^{39} -4.86420 q^{41} -5.24127 q^{43} +16.8798 q^{45} +5.11346 q^{47} -5.19100 q^{49} -2.90427 q^{51} -3.63971 q^{53} -2.30747 q^{55} +3.21174 q^{57} -13.7786 q^{59} +3.90408 q^{61} +9.83897 q^{63} -3.10353 q^{65} +5.72084 q^{67} +1.76303 q^{69} +4.49402 q^{71} +1.28228 q^{73} -1.04201 q^{75} -1.34499 q^{77} +6.80150 q^{79} +22.5674 q^{81} +7.06560 q^{83} +2.08657 q^{85} +13.9005 q^{87} +15.8923 q^{89} -1.80900 q^{91} +6.08279 q^{93} -2.30747 q^{95} +12.2842 q^{97} -7.31527 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 4 q^{3} + q^{5} + 4 q^{9} - 6 q^{11} - 7 q^{15} + 3 q^{17} - 6 q^{19} + 7 q^{21} - 7 q^{23} + 3 q^{25} - 13 q^{27} - 4 q^{29} - 7 q^{31} + 4 q^{33} - 6 q^{35} - 2 q^{37} - 7 q^{39} + 7 q^{41} - 21 q^{43}+ \cdots - 4 q^{99}+O(q^{100}) \) 6 * q - 4 * q^3 + q^5 + 4 * q^9 - 6 * q^11 - 7 * q^15 + 3 * q^17 - 6 * q^19 + 7 * q^21 - 7 * q^23 + 3 * q^25 - 13 * q^27 - 4 * q^29 - 7 * q^31 + 4 * q^33 - 6 * q^35 - 2 * q^37 - 7 * q^39 + 7 * q^41 - 21 * q^43 + 11 * q^45 - 16 * q^47 + 2 * q^49 - 15 * q^51 + 17 * q^53 - q^55 + 4 * q^57 - 19 * q^59 - 6 * q^61 - 2 * q^63 + 6 * q^65 - 14 * q^67 + q^69 - q^71 - 5 * q^73 - 18 * q^75 + 2 * q^81 - 11 * q^83 - 26 * q^85 - 14 * q^87 + 12 * q^89 - 44 * q^91 - 6 * q^93 - q^95 + 14 * q^97 - 4 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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